teaching, math, teaching math

What Is Mathematics?

On Friday at PCMI, Jonathan Mattingly gave an afternoon lecture entitled “Math and Gerrymandering.” You can see the talk here, and explore Jonathan and his team’s great work here. It was a fascinating talk, and I learned a ton. Jonathan spoke about Supreme Court cases on gerrymandering that have been recently decided or could be decided soon, looked at the statistics behind how his team models gerrymandering, and explored some really cool representations of gerrymandering to help understand what it means from a less technical perspective. One interesting point is that gerrymandering is not just about oddly-shaped districts. The two maps below are equally biased toward Republicans.

While oddly shaped districts may be a symptom, they could also be bringing together an interest group that wouldn’t have representation otherwise. Mattingly points to something else as what he calls the “signature of gerrymandering,” shown in the image below and which he writes about here:

See how the orange and purple lines jump between 10 and 11 districts, but the yellow and green ones increase more uniformly? The jump means that, in gerrymandered maps, Democrats could get anywhere from 50%-65% of the vote, and their representation in the House of Representatives wouldn’t change. As a contrast, the green data points are a hypothetical map that a non-partisan commission drew but did not implement. In that map, more votes for one side correlate strongly with more seats in the House, and there’s no large jump indicating an advantage for one side

It was a great talk, but one piece of it bugged me. Both Mattingly and Rafe Mazzeo, the director of PCMI who introduced the talk, alluded to the idea that his project is “social” work and not actually mathematics. The first part of the talk I mostly understood, but toward the end he put up a slide that said “Where’s the Mathematics?” (implying that communicating about gerrymandering isn’t mathematics) and then launched into something I didn’t understand and that also seemed much less relevant to understanding and communicating about gerrymandering to the public.

I don’t mean to attack Jonathan, but I see a missed opportunity for expanding the meaning of mathematics here. This bugged me, and I asked a question at the end of the talk trying to push on this. Here is a more articulate version of that question:

In your talk, you alluded several times to the idea that studying gerrymandering is not “real mathematics.” I would conjecture that many who are in or consider entering the field of mathematics would agree — we often see mathematics as Lie groups, Fourier series, eight-dimensional topology, and other stuff I don’t understand. But many others would argue that studying gerrymandering, and applying mathematics to social and political questions more broadly, is mathematics. More than that, social and political work has the potential to change what we perceive as mathematics, and transform our discipline into one that can more concretely improve the condition of humanity. What would you say to a high school student, undergraduate, or graduate student who sees your work as “not mathematics,” and how can we work to create a more inclusive mathematics in the future?

I’d like to go a step further. Most of my students don’t see themselves as mathematicians because they can’t see pathways for mathematics to positively influence their lives. What if, as one small step toward creating richer perceptions of what mathematics is and creating a discipline that has a more positive influence on humans, we chose to center “mathematics for social good” as a core part of what we see as math?

What if university math departments designed undergraduate courses on the mathematics of, say, gerrymandering, income inequality, and applications of big data. What if those departments required math majors to take at least one of these courses, under the umbrella of “mathematics for social good” as part of a math major?

What if high schools designed a pathway focusing on similar themes as an alternative to the current race to calculus?

What if professionals prioritized communication with the public about mathematics as equal in stature to proving theorems?

Michael Pershan wrote recently about how mathematics was taught several hundred years ago — to very briefly summarize, undergraduates read texts like Euclid’s Elements and had to be able to state proofs or theorems from the text or solve simple problems orally. There was very little written mathematics, and very little of what we might call problem solving. And, to them, that was the discipline of mathematics. Universities didn’t change until the middle of the 19th century. That blows my mind. The way we look at what is important in mathematics education has radically changed in the last two hundred years. Why can’t it change again?

I’m interested in conceptualizing “mathematics for social good” because it’s on my mind at the moment, but more broadly, I’m curious what we might change if we had the opportunity to completely reimagine mathematics education. What essential things do we want people to know? What experiences do we want people to have? How do we want the public to understand mathematics? I think those questions might cause us to build something that looks very different than the mathematics we have today.

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7 thoughts on “What Is Mathematics?”

Another question I’ve been wondering lately is if mathematics can change, what does it mean for mathematics to change? In other words, maybe there is no “mathematics,” and we can just do whatever you want, and aren’t accountable to anything called “mathematics” at all.